This will:

#!/usr/bin/perl
use strict;
use warnings;
use Data::Dumper;
use diagnostics;

undef $/;
my $text = <DATA>;

my @matches;
if (@matches = $text =~ /alpha\|([^|]+)\|/smg) {
    foreach my $match (@matches) {
        print "match: $match\n";
    }
}

__DATA__
alpha|beta|alpha|theta|alpha|gamma|alpha|episilon
[download]

Output:

match: beta
match: theta
match: gamma
[download]

Or you could dispense with loading data into the array at all (unless you need it for later):

#!/usr/bin/perl
use strict;
use warnings;
use Data::Dumper;
use diagnostics;

undef $/;
my $text = <DATA>;

while ($text =~ /alpha\|([^|]+)\|/smg) {
    print "match: $1\n";
}

__DATA__
alpha|beta|alpha|theta|alpha|gamma|alpha|episilon
[download]